508 Captions

508 CAPTIONS

Figures

This graph plots six scenarios to compare wind velocity and damping relation of inclined dry cable. The horizontal axis is the Scruton Number (S subscript C equals M (mass of cable per unit length) times zeta (damping ratio) divided by rho (air density) times D (cable diameter) squared) ranging from 0 to 18. The vertical axis is reduced wind velocity (U lowercase R) ranging from 0 to 360. Saito’s rig test data defines the boundary for instability for rain/wind oscillations shown as a rising solid line from 45 to 105 velocities and a Scruton number of approximately 8, which is where Saito equals theta at 45 degrees. The FHWA instability line coincides with its small amplitude of less than 10 millimeters (0.39 inch). Miyata and both FHWA lines from 10 to 80 millimeters (0.39 to 3.1 inches) and maximum 80 millimeters (3.1 inches) all plot in the unstable range below a Scruton number of 3. The data suggest that rather than the recommended Scruton number of 10 to suppress rain/wind vibrations that low structural damping allows a Scruton number as low as 3.

This plot is based on the computer model of the Rama 8 Bridge in Bangkok, Thailand. The third longest cable (M26) was studied to determine the effects of live loading. The horizontal axis is time in seconds ranging from 0 to 40. The vertical axis is tension in kilonewtons. The straight green line is the minimum static load at 0 and the straight blue line is the maximum static load at 225 kilonewtons (50,625 poundforce). The static and dynamic effects of the transit train vary from negative 25 at 8 seconds to a high of 230 kilonewtons (51,750 poundforce). The increase in maximum cable tension from dynamic effects is less than 10 percent. (1 kilonewton equals 225 poundforce.)

Figure 3. Graph. Time history and power spectral density (PSD) of the first 2 hertz for deck at midspan (vertical direction).

This figure plots measurements of deck and stay movements in a vertical direction at the Fred Hartman Bridge during a storm. The first plot shows time on the horizontal axis from 0 to 5 minutes. The vertical axis is acceleration time history for cable AS24. With a natural frequency of 0.59 hertz, the cable oscillation expands to negative 2.5 to almost 3 hertz beginning at minute 3.5. The second plot shows a horizontal axis of frequency in hertz ranging from 0 to 2. The vertical axis is the power spectral density of the deck at midspan (G squared divided by hertz) ranging from 0 to 0.15. The power spectral density of the bridge deck spikes to 0.1 at 0.6 hertz.

Figure 4. Graph. Time history and power spectral density (PSD) of the first 2 hertz for cable at AS24 (in-plane direction) deck level wind speed.

This figure plots measurements of both deck and stay movements (in-plane direction) at the Fred Hartman Bridge during a storm. The first horizontal axis is time from 0 to 5 minutes. The first vertical axis is acceleration time history for the deck at midspan ranging from negative 0.2 to 0.2. At 2 minutes, the deck begins to oscillate up to negative 0.1 to 0.1 until 5 minutes. The second plot shows a horizontal axis of frequency in hertz ranging from 0 to 2. The vertical axis is the power spectral density of cable AS24 (G squared divided by hertz) ranging from 0 to 0.15. The power spectral density of the cable shows three spikes: one rising to 0.13 at 0.6 hertz, a minimal one at 1.1 hertz and a PSD of 0.15 at 1.7 hertz.

Figure 5. Graph. Deck level wind speed.

The graph shows the wind speed time history at the deck of the Fred Hartman Bridge during a storm. The horizontal axis is time in minutes ranging from 0 to 5 minutes. The vertical axis is wind speed in meters per second, ranging from 8 to 11.5. The wind starts at 8.5 and increases to 11.5 meters per second over 5 minutes.

Figure 6. Photo. Damper at cable anchorage.

The photo shows a close view of a damper at a cable anchorage. The damper is a black and blue neoprene viscous circle that fits around the cable to absorb and minimize vibration. The white cylinders are the cable surface treatment.

Figure 7. Drawing. Taut cable with a linear damper.

This diagram is used to illustrate the dynamics of a taut cable-damper system. T represents the tension of the taut cable, L represents the cable length, lowercase L equals the distance of the damper from a cable end, lowercase M is the mass per unit length of the cable, lowercase C is the damping coefficient of the damper, and lowercase X represents the distance along the cable axis from one end of the cable.

In this graph, the horizontal axis is kappa, the nondimensional damping parameter or the range of normalized damping coefficients from 0 to 1 in 10th increments. The vertical axis is the damping ratio ranging from 0 to 0.6 for the first five modes of damper location 0.02. The curves all coincide with 0.50 the optimum damping ratio for a linear damper for one mode of vibration.

This graph shows the dynamic behavior of a taut cable with a passive, nonlinear, power-law damper attached at an intermediate point. Both the expression for kappa, the horizontal axis ranging from 0 to 1 and the damping ratio of the vertical axis ranging from 0 to 0.6 is the same as the previous graph. The value of beta (the damping exponent) at 0.5 changes the shape of the curve slightly, but the curve is nearly invariant with damper location and mode number over the same range of parameters as the estimation curve for the linear damper. This special case of a square root damper shows performance is independent of mode number and depends only on the vibration amplitude.

Figure 10. Photo. Fred Hartman Bridge.

The photo shows the twin-deck cable-stayed four-lane bridge over the Houston ship channel. The bridge has a central span of 380 meters (1,250 feet) with 192 cables in four inclined planes, spaced at 15-meter (50-foot) intervals.

Figure 11. Photo. Cable crosstie systems.

Crossties increase the in-plane stiffness of bridge stays. The picture shows eight cables joined by a cable crosstie system, a set of secondary transverse cables. The dynamic properties of single cables are modified by such lateral constraints into a more complex cable network. Such a system works by increasing the generalized mass and the Scruton number in the lower modes and raising the frequencies and localizing the vibration in higher modes and is likely to increase the effective damping of cable through energy transfer between cables.

Figure 12. Photo. Dames Point Bridge.

This bridge built in 1989, located in Jacksonville, Florida, is an example of an older bridge retrofitted with crossties before the discovery of rain/wind vibrations. No problematic cable vibrations have been reported on this bridge.

Figure 13. Chart. General problem formulation.

The diagram shows a simplified network with a set of two taut cables (T subscript 1 and T subscript 2 cable tension) that are L subscript 1 and L subscript 2 long, connected by a vertical rigid rod (P subscript 1 and P subscript 2). Cable tensions are divided into Elements 1,1; 1,2 and 2,1; 2,2. The location of the vertical rigid connection crosstie is represented by I subscript 1 and I subscript 2. The horizontal and vertical coordinates are represented by X and Y, respectively, with subscripts 11, 12, 21, and 22 representing the origin located at the left end of the top cable, the right end of the top cable, the left end of the bottom cable, and the right end of the bottom cable, respectively.

The free-vibration analysis method was used to study a cable network modeled after the Fred Hartman Bridge. The main south tower span unit shows a set of 12 stays, numbered from 13S to 24S with the latter as the reference element. The three transverse connectors (an eight-loop steel wire rope system) labeled Restraint 1 through 3 are spaced about one-fourth the distance along the reference cable 24S.

The eigenfunctions for the model show that along a normalized abscissa, mode 5 shows an uneven amplitude pattern influenced by the location of the transverse connectors, governed by the distance between consecutive connectors.

This graph shows the natural frequencies plotted as a function of the mode number and compared to individual cable behavior. The horizontal axis is the mode number from 0 to 35. The vertical axis is frequency in hertz, ranging from 0.5 to 4.5. The lower limit of the plateau frequency interval is 1.9, while the upper limit is 2.7 hertz. The graph shows a high density pattern from 0 to 5 modes up to 4.5 hertz frequency for the localized modes. The three comparative cases for network vibration are the original configuration, rigid transverse links and modified non-rigid configuration with ground restrainers that essentially overlap in the graph. The consecutive step pattern changes at the lower and upper frequency interval plateaus (1.9 and 2.7 hertz).

The graphic shows the configuration for the field tests on the south tower of the Fred Hartman Bridge with three crosstie restraints and cables AS1 though AS12. The accelerometers are at a height of 6.7 meters (22 feet) placed along the stays on cables AS1, AS3, AS5, and AS9.

Figure 19. Drawing. Types of cable surface treatments.

The diagram shows three types of cable surface treatment to mitigate rain/wind vibrations. They include the lumped surface roughness that provides an uneven cable to disrupt rivulets, the axially aligned protrusions that occur every 30 degrees that are 5 millimeters (0.2 inch) high and 11 millimeters (0.43 inch) wide, and the helical fillets that have a 60-millimeter (2.34-inch) protrusion to form a double helix spiral bead formation, common on new bridges.

Test data for spiral bead cable surface treatment show how the helical filet reduces vibration. The horizontal axis is wind velocity ranging from 5 meters per second (11 miles per hour) to 15 meters per second (33 miles per hour). Without the filet, the vibration amplitude can rise to as much as 1 meter (3.3 feet) while with the filet the highest reading is 0.05 meter (1.6 feet).

Figure 21. Photo. Leonard P. Zakim Bunker Hill Bridge.

The picture shows the bridge over the Charles River in Boston, Massachusetts, with cable-stayed inverted-Y shaped towers and a main span of 227 meters (745 feet). The bridge is 56 meters (183 feet wide) with 10 lanes (2 cantilevered). The ungrouted cables are arranged in a single plane for the back spans and two inclined planes for the main span. Cable vibration mitigation measures include external visco-elastic dampers at the roadway level, cable crossties, and a double helical fillet cable treatment.

Figure 22. Graph. Sample decay: No damping and no crossties.

The graph shows a time history of decay in manually excited cables with neither damping nor crossties. The time ranges from 0 to 2 times 10 to the fourth power. The graph plots six accelerometer boxes in three dimensions (X, Y, and Z). The raw signal ranges from a low of negative 0.5 to a high of near 6.

Figure 23. Graph. Sample decay: With damping and no crossties.

The graph shows a time history of decay in manually excited cables with damping only. The time ranges from 0 to 2 times 10 to the fourth power. The graph plots six accelerometer boxes in three dimensions (X, Y, and Z). The raw signal ranges from a low of negative 1.9 to a high of near 5. This graph shows an improvement in vibration compared to no mitigation measures.

Figure 24. Graph. Sample decay: With damping and crossties.

The graph shows a time history of decay in manually excited cables. The time ranges from 0 to 2 times 10 to the fourth power. The raw signal ranges from a low of negative 2.0 to a high of 6. The graph plots six accelerometer boxes in three dimensions (X, Y, and Z). While the range is wider, this configuration with both damping and crossties performs the best because the amplitudes are short and not sustained.

Figure 25. Photo. Sunshine Skyway Bridge.

This bridge has been in service since 1982 with no cable vibration problems. The bridge spans 366 meters (1,200 feet) with 2 single-mast towers and 84 cables in a single plane. The cables only have external viscous dampers.

Figure 26. Photo. Stay and damper brace configuration.

This closeup view shows the single mast tower with 21 stays radiating from it and the damper brace configuration on every stay anchored to the bridge deck. Two inclined struts are each connected to three viscous dampers for damping in two directions. Even without crossties or cable surface treatment, this bridge reports no vibration problems.

Figure 27. Photo. Reference database search page.

The Microsoft® database for cable stay vibration references includes search entries for article titles, authors, reference information, and abstracts when available. The orange buttons in the left margin allow a user to sign up, search, enter data, and return to home. The database can have a timeframe by year and be sorted by article name, journal name, or date.

Figure 28. Photo. Reference database search results page.

The Microsoft format search results for articles show the number of results that match criteria and then a numbered list with the title, authors, source and date, volume, issue, and pages with an abstract if available. The civil engineering department at John Hopkins University developed the database.

Figure 29. Photo. U.S. cable-stayed bridge database: Switchboard.

The user selects a switchboard that controls data entry and referential integrity between categories which includes general bridge information, wind data and cable data. Users can edit or make a new entry in each category. The example shows the cursor set on general bridge information and the PR 148 over La Plata River.

The database provides two tabs under general bridge information, general and structural information. Under the general information are the bridge name, the NBIS number, the owner, the designer, the wind consultant, the cable supplier, the year the design was completed, northing, year the construction was complete, easting, if global dynamics are available, the alignment, and the units for data entry along with photo figures.

Figure 31. Photo. U.S. cable-stayed bridge database: Cable data.

The cable data screen has four information tabs: cable geometry, cable properties (strand type, size dead load tension and the protection system), cable connections (upper and lower anchors and fatigue test information), and aerodynamic details (dampers and sheathing surface treatment). Information for each cable of the bridge is provided in a separate screen. Under cable geometry, each cable number can be detailed with the upper and lower coordinates (X, Y, and Z) and the cable length. A go button lets the user calculate cable length. A pulldown menu lets the user select a cable number.

Figure 32. Photo. U.S. cable-stayed bridge database: Wind data.

The wind data screen has two tabs for design data and vibration measurements. Under design data, the user can access wind climatology studies if available, and view the superstructure mass, the design wind speeds with the nearest recording station and the vibration measurement information (mode number, type, and frequency in hertz).

Figure 33. Graph. Galloping of inclined cables.

The graph shows a solid ascending curved line for Saito’s 1994 data derived from wind tunnel experiments plotted in the form of the formula for critical wind speed that causes galloping (0 to 644 kilometers per hour (0 to 400 miles per hour)) on the vertical axis and Scruton numbers on the horizontal axis ranging from 0 to 20. The dotted lines are C values, constants from 25 to 55 from Cooper’s global stability criteria. Close cable spacings are in the 25 range while the 10 to 20 diameter range is about 80. All data points except one fall above the curve for C equals 40, so that value can be used to predict single inclined cable galloping.

Figure 34. Drawing. Aerodynamic devices.

The diagram shows three types of cable surface treatment to mitigate rain/wind vibrations. They include the lumped surface roughness that provides an uneven cable to disrupt rivulets, the axially aligned protrusions that occur every 30 degrees that are 5 millimeters (0.2 inch) high and 11 millimeters (0.43 inch) wide, and the helical fillets that have a 60-millimeter (2.34-inch) protrusion to form a double helix spiral bead formation, common on new bridges.

Figure 35. Drawing. Cable crossties.

The diagram shows a bridge cross section. The tower has a set of 17 cable stays extending from each side. Three crossties connect from the bridge deck to the outermost cables to raise the natural frequency of the cables because the transverse array changes the effective cable length.

Figure 36. Drawing. Viscous damping.

The cable stay is connected on an angle to the bridge deck. The viscous (oil) damper attached to the cable at a right angle completes a triangle when attached to the bridge deck. Damping of long cables tends to be low, so adding small amounts at or near the cable ends dramatically improves aerodynamic stability.

Figure 37. Drawing. Material damping.

Material damping has been used successfully to improve aerodynamics on existing bridges. A damping ratio of 0.5 percent is a minimum threshold to reduce cable instability. Damping methods include petroleum wax infill in the guide pipes flush to the bridge deck, neoprene bushings at the cable anchorages, or visco-elastic dampers in the cable anchorage pipe.

The diagram shows X as the horizontal bridge axis, Z as the vertical axis, and Y as the depth. The orientation of the cable in relation to the mean wind direction can be represented by two angles. The vertical inclination angle theta represents the angle of the cable in a vertical plane parallel to the bridge axis. If the wind bows with a horizontal angle of beta to the bridge axis, the projection of the cable on the horizontal plane makes a horizontal yaw angle of beta with the wind vector. The vector OB represents the direction of cable motion normal to the wind direction and cable axis and has an angle of alpha in relation to this horizontal projection. These relationships can be represented by trigonometric formulas.

Figure 39. Photo. Cable supporting rig: Top.

The picture illustrates the top spring rig in the wind tunnel (the top and bottom rigs are each set perpendicular to the cable axis). The top rig view shows where the red metal supporting structure connects the springs that are perpendicular to the cable. The green wind tunnel sides can be seen in the background. The two pairs of springs (top and bottom) at both ends can be rotated around the central axis of the cable models.

Figure 40. Photo. Cable supporting rig: Bottom.

The picture illustrates the bottom spring rig in the wind tunnel (the top and bottom rigs are each set perpendicular to the cable axis). The bottom spring rig shows all four springs underneath the platform that holds the cable and connected to the side walls of the wind tunnel. The two pairs of springs (top and bottom) at both ends can be rotated around the central axis of the cable models.

The figure shows the test wind tunnel, an open circuit blowing type with a 7.93-meter (26-foot) diameter fan at the entry. The air flow enters at the inlet screen into the contraction cone and accelerates the flow with the straightening vanes, vortex generators and motor drive. Screens help make the flow velocity uniform. The removable model access cover in the roof is 3.7 meters (12 feet long) to install the model. The elevating tunnel floor can be raised 0.46 meters (1.5 feet) to work on installation or simulate varying ground effects. The wind tunnel working section is 12 meters (39.36 feet) long, 3 meters (9.84 feet) wide, and 6 meters (19.68 feet) high. Wind is released through the vaned elbow diffuser and the silencer.

Figure 42. Drawing. Cross section of the working section of propulsion wind tunnel.

The cross section shows the air supply pipe and the air bearing seal that enter the wind tunnel. The floor drive mechanism and the floor jacking screw are the details that allow the tunnel floor to be variable heights. The working section of the wind tunnel is 3.05 meters (10 feet) wide and 6.10 meters (20 feet) high.

Figure 43. Photo. Data acquisition system.

The picture shows the room where data acquisition from the wind tunnel took place. The console has 7 input channels to read the temperature, downwind, and upwind direction displacement for X and Y, and transducers to read the pressure from the Pitot tubes on the tunnel walls. Video display terminals allow for real-time viewing of the laser sensors in the tunnel. The monitors linked to the video cameras record instability of cable motion under different wind velocities. The data was saved in both time history (.TMS) and statistical summary files (.STA).

Figure 44. Photo. Airpot damper.

The damper is a black low-friction cylindrical tube with a protruding connecting rod at the top and the mounting stud with the red adjustable orifice at the bottom.

Figure 45. Drawing. Cross section of airpot damper.

The cross section shows the airpot damper with the mounting stud at the bottom near the vent path and the adjustable orifice. The cylinder contains a graphite carbon piston, and ball joint installed with a connecting rod.

Figure 46. Photo. Elastic bands on the spring coils.

When the airpot dampers had significant impact on the small amplitude motion, they were used for only very high level damping. For intermediate and high level damping cases, elastic bands were substituted along the spring coils to increase damping. The picture shows the top spring coil rig with elastic bands applied in four-coil increments. The desired damping levels were achieved by adjusting the number and locations of the elastic bands.

Figure 47. Drawing. Side view of setups 1B and 1C.

The drawing shows the cable in the wind tunnel with the cable 6,700 millimeters (261 inches) long. The model cable angle phi is set at 45 degrees to the horizontal and the adjustable floor height is 138 millimeters (5.38 inches). The wind velocity (U) comes from the left to vibrate the cable.

Figure 48. Drawing. Side view of setups 2A and 2C.

The drawing shows the cable in the wind tunnel with the cable 6,700 millimeters (261 inches) long. The model cable angle phi is set at 60 degrees to the horizontal and the adjustable floor height is zero. The drawing shows the height of the wall where the cable is attached to be 6,000 millimeters (234 inches). The wind velocity (U) comes from the left to vibrate the cable.

Figure 49. Drawing. Side view of setups 3A and 3C.

The drawing shows the cable in the wind tunnel with the cable 6,700 millimeters (261 inches) long. The model cable angle phi is set at 30 degrees to the horizontal and the adjustable floor height is 233 millimeters (9.1 inches). The wind velocity (U) comes from the left to vibrate the cable.

Figure 50. Photo. Cable setup in wind tunnel for testing.

The photo shows the long view of the wind tunnel with a cable set up attached to the ceiling and the bottom spring mechanism. Light sources include the side windows and the inlet screen at the contraction cone end.

This figure shows graphs for the sway (horizontal) and vertical amplitude for setup 2C (smooth surface cable with low damping) and a vertical angle theta at 60 degrees. The horizontal axis is the sway or vertical amplitude in millimeters created by hand excitation from 0 to 35 millimeters (0 to 1.4 inches). The vertical axis is the damping ratio from 0.00 to 0.12. For the sway amplitude, the points cluster at 5 to 15 millimeters (0.2 to 0.6 inch) in a range from 0.04 to 0.06 percent damping ratio. The vertical amplitude shows the point cluster in a lower range of 2 to 10 millimeters (0.08 to 0.4 inch) at 0.02 to 0.06 damping ratio.

This graph compares the divergent response of the sway and vertical amplitude for setup 2C. The horizontal axis is the wind velocity (U) in meters per second ranging from 0 to 40. The vertical axis is amplitude (the minimum subtracted from the maximum and divided by 2 to derive an average) in millimeters. For both parameters, the amplitude remains below 20 millimeters (0.8 inch) from 10 to 30 meters per second (32.8 to 98.4 feet per second). When the wind increases to 33 meters per second (108.2 feet per second), the sway rises to 22, while the vertical amplitude increases markedly from 25 to 80 millimeters (1 to 3 inches).

Figure 53. Graph. Lower end X-motion, time history of setup 2C at U = 32 meters per second (105 feet per second).

The graph shows the lower end X-motion displacement for setup 2C with the wind velocity at 32 meters per second (105 feet per second). The horizontal axis is time in seconds ranging from 0 to 150. The vertical axis is X-displacement at the lower end in millimeters ranging from negative 100 to 100. The displacement increases from 15 millimeters (0.6 inch) at 0 to negative 20 and 30 millimeters (negative 0.8 and 1.2 inches) at 150 seconds.

Figure 54. Graph. Top end X-motion, time history of setup 2C at U = 32 meters per second (105 feet per second).

The graph shows the top end X-motion displacement for setup 2C with the wind velocity at 32 meters per second (105 feet per second). The horizontal axis is time in seconds ranging from 0 to 150. The vertical axis is X-displacement at the top end in millimeters ranging from negative 100 to 100. The displacement increases from negative 7 and 10 millimeters (0.3 and 0.4 inch) at 0 to negative 20 and 25 millimeters (negative 0.8 and 1 inch) at 150 seconds.

Figure 55. Graph. Lower end Y-motion, time history of setup 2C at U = 32 meters per second (105 feet per second).

The graph shows the lower end Y-motion displacement for setup 2C with the wind velocity at 32 meters per second (105 feet per second). The horizontal axis is time in seconds ranging from 0 to 150. The vertical axis is X-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 and 3.9 inches). The displacement increases from negative 40 and 35 millimeters (negative 1.6 and 1.4 inches) at 0 to negative 85 and 75 millimeters (negative 3.3 and 2.9 inches) at 150 seconds.

Figure 56. Graph. Top end Y-motion, time history of setup 2C at U = 32 meters per second (105 feet per second).

The graph shows the top end Y-motion displacement for setup 2C with the wind velocity at 32 meters per second (105 feet per second). The horizontal axis is time in seconds ranging from 0 to 150. The vertical axis is X-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement increases from negative 40 and 35 (negative 1.6 and 1.4 inches) at 0 to negative 80 and 70 millimeters (negative 3.1 and 2.7 inches) at 150 seconds.

The graph shows the trajectory of setup 2C with wind velocity at 32 meters per second (105 feet per second). The horizontal axis is the mean X-displacement in millimeters ranging from negative 200 to 200 (negative 7.8 to 7.8 inches) while the vertical axis is the mean Y-displacement ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The plot shows an upright ellipse centered around 0 with the X-displacement around negative 20 to 20 millimeters (negative 0.8 to 0.8 inches) and the larger Y-displacement at negative 80 to 75 millimeters (negative 3.1 to 2.9 inches).

Figure 58. Graph. Lower end X-motion, time history of setup 2A at U = 18 meters per second (59 feet per second) in the first 5 minutes.

The graph shows the lower end X-motion displacement for setup 2A with the wind velocity at 18 meters per second (59 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is X-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement increases from negative 20 and 15 millimeters (negative 0.8 and 0.6 inch) at 0 to negative 55 and 50 millimeters (negative 2.1 and 2.0 inches) at 300 seconds.

Figure 59. Graph. Top end X-motion, time history of setup 2A at U = 18 meters per second (59 feet per second) in the first 5 minutes.

The graph shows the top end X-motion displacement for setup 2A with the wind velocity at 18 meters per second (59 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is X-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement increases from negative 20 and 15 millimeters (negative 0.6 and 0.8 inch) at 0 to negative 50 and 50 millimeters (negative 2 to 2 inches) at 300 seconds.

Figure 60. Graph. Lower end Y-motion, time history of setup 2A at U = 18 meters per second (59 feet per second) in the first 5 minutes.

The graph shows the lower end Y-motion displacement for setup 2A with the wind velocity at 18 meters per second (59 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is Y-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement varies minimally in a straight line from negative 7 millimeters (negative 0.3 inch) at 0 to negative 2 and negative 10 millimeters (negative 0.08 to 0.4 inch) at 300 seconds.

Figure 61. Graph. Top end Y-motion, time history of setup 2A at U = 18 meters per second (59 feet per second) in the first 5 minutes.

The graph shows the top end Y-motion displacement for setup 2A with the wind velocity at 18 meters per second (59 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is Y-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement varies minimally in a straight line from negative 7 millimeters (negative 0.3 inch) at 0 to negative 2 and negative 10 millimeters (negative 0.08 to 0.4 inch) at 300 seconds.

Figure 62. Graph. Lower end X-motion, time history of setup 2A at U = 18 meters per second (59 feet per second) in second 5 minutes.

The graph shows the lower end X-motion displacement for setup 2A with the wind velocity at 18 meters per second (59 feet per second) in the second five minutes. The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is X-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement remains nearly steady at negative 60 and 55 millimeters (negative 2.3 and 2.1 inches) at 0 and the same values at 300 seconds.

Figure 63. Graph. Top end X-motion, time history of setup 2A at U = 18 meters per second (59 feet per second) in second 5 minutes.

The graph shows the top end X-motion displacement for setup 2A with the wind velocity at 18 meters per second (59 feet per second) in the second five minutes. The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is X-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement remains nearly steady at negative 55 and 50 (negative 2.3 and 2 inches) at 0 and the same values at 300 seconds.

Figure 64. Graph. Lower end Y-motion, time history of setup 2A at U = 18 meters per second (59 feet per second) in second 5 minutes.

The graph shows the lower end Y-motion displacement for setup 2A with the wind velocity at 18 meters per second (59 feet per second) in the second five minutes. The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is Y-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement varies minimally in a straight line from negative 7 millimeters (negative 0.3 inch) at 0 to negative 2 and negative 10 millimeters (negative 0.08 to 0.4 inch) at 300 seconds.

Figure 65. Graph. Top end Y-motion, time history of setup 2A at U = 18 meters per second (59 feet per second) in second 5 minutes.

The graph shows the top end Y-motion displacement for setup 2A with the wind velocity at 18 meters per second (59 feet per second) in the second five minutes. The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is Y-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement varies minimally in a straight line from negative 7 millimeters (negative 0.3 inch) at 0 to negative 2 and negative 10 millimeters (negative 0.08 to 0.4 inch) at 300 seconds.

Figure 66. Graph. Lower end X-motion, time history of setup 2A at U equals 19 meters per second (62 feet per second).

The graph shows the lower end X-motion displacement for setup 2A with the wind velocity at 19 meters per second (62 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is X-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement decreases from negative 80 and 70 millimeters (negative 3.1 and 2.7 inches) at 0 to negative 55 and 50 millimeters (negative 2.1 and 2 inches) at 300 seconds.

Figure 67. Graph. Top end X-motion, time history of setup 2A at U equals 19 meters per second (62 feet per second).

The graph shows the top end X-motion displacement for setup 2A with the wind velocity at 19 meters per second (62 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is X-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement decreases from negative 70 and 60 millimeters (negative 2.7 and 2.3) at 0 to negative 50 and 50 millimeters (negative 2 and 2 inches) at 300 seconds.

Figure 68. Graph. Lower end Y-motion, time history of setup 2A at U equals 19 meters per second (62 feet per second).

The graph shows the lower end Y-motion displacement for setup 2A with the wind velocity at 19 meters per second (62 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is Y-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement varies minimally in a straight line from negative 7 millimeters (negative 0.3 inch) at 0 to negative 2 and negative 10 millimeters (negative 0.08 to 0.4 inch) at 300 seconds.

Figure 69. Graph. Top end Y-motion, time history of setup 2A at U equals 19 meters per second (62 feet per second).

The graph shows the top end Y-motion displacement for setup 2A with the wind velocity at 19 meters per second (62 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is Y-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement varies minimally in a straight line from negative 7 millimeters (negative 0.3 inch) at 0 to negative 2 and negative 10 millimeters (negative 0.08 to 0.4 inch) at 300 seconds.

Figure 70. Graph. Lower end X-motion, time history of setup 1B at U equals 24 meters per second (79 feet per second).

The graph shows the lower end X-motion displacement for setup 1B with the wind velocity at 24 meters per second (79 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is X-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement ranges from negative 20 to 20 millimeters (negative 0.8 and 0.8 inch) at 0 to negative 30 and 30 millimeters (negative 1.2 and 1.2 inch) at 300 seconds.

Figure 71. Graph. Top end X-motion, time history of setup 1B at U equals 24 meters per second (79 feet per second).

The graph shows the top end X-motion displacement for setup 1B with the wind velocity at 24 meters per second (79 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is X-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement ranges from negative 20 to 20 millimeters (negative 0.8 and 0.8 inch) at 0 to negative 25 and 30 millimeters (negative 1.0 and 1.2 inch) at 300 seconds.

Figure 72. Graph. Lower end Y-motion, time history of setup 1B at U equals 24 meters per second (79 feet per second).

The graph shows the lower end Y-motion displacement for setup 1B with the wind velocity at 24 meters per second (79 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is Y-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement varies minimally in a straight line from 5 to 7 millimeters (0.2 to 0.3 inch) at 0 and the same values at 300 seconds.

Figure 73. Graph. Top end Y-motion, time history of setup 1B at U equals 24 meters per second (79 feet per second).

The graph shows the top end Y-motion displacement for setup 1B with the wind velocity at 24 meters per second (79 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is Y-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement has no variation and reads in a straight line at about 5 millimeters (0.2 inch) from 0 to 300 seconds.

Figure 74. Graph. Lower end X-motion, time history of setup 1C at U equals 36 meters per second (118 feet per second).

The graph shows the lower end X-motion displacement for setup 1C with the wind velocity at 36 meters per second (118 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is X-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement ranges from 10 to 20 millimeters (0.4 to 0.8 inch) at 0, peaks at 0 to 20 millimeters (0 to 0.8 inch) at 90 seconds and ends at 10 to 15 millimeters (0.4 to 0.6 inch) at 300 seconds.

Figure 75. Graph. Top end X-motion, time history of setup 1C at U equals 36 meters per second (118 feet per second).

The graph shows the top end X-motion displacement for setup 1C with the wind velocity at 36 meters per second (118 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is X-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement ranges from 0 to 15 millimeters (0 to 0.6 inch) at 0 with a peak of negative 5 to near 20 millimeters (negative 0.2 to 0.8 inch) at 90 seconds and ranges from 2 to 10 millimeters (0.08 to 0.4 inch) at 300 seconds.

Figure 76. Graph. Lower end Y-motion, time history of setup 1C at U equals 36 meters per second (118 feet per second).

The graph shows the lower end Y-motion displacement for setup 1C with the wind velocity at 36 meters per second (118 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is Y-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement varies minimally from negative 30 to 30 millimeters (negative 1.2 to 1.2 inches) at 0 to negative 20 to 25 millimeters (negative 0.8 to 1.0 inch) at 300 seconds.

Figure 77. Graph. Top end Y-motion, time history of setup 1C at U equals 36 meters per second (118 feet per second).

The graph shows the top end Y-motion displacement for setup 1C with the wind velocity at 36 meters per second (118 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is Y-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement varies minimally from negative 20 and 25 millimeters (negative 0.8 to 1.0 inch) at 0 to negative 15 and negative 22 millimeters (negative 0.6 and negative 0.9 inch) at 300 seconds.

Figure 78. Lower end X-motion, time history of setup 3A at U equals 22 meters per second (72 feet per second). Graph.

The graph shows the lower end X-motion displacement for setup 3A with the wind velocity at 22 meters per second (72 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is X-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement remains almost constant from negative 20 to 20 millimeters (negative 0.8 to 0.8 inch) at 0 to the same values at 300 seconds.

Figure 79. Top end X-motion, time history of setup 3A at U equals 22 meters per second (72 feet per second). Graph.

The graph shows the top end X-motion displacement for setup 3A with the wind velocity at 22 meters per second (72 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is X-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement varies minimally from negative 15 to 15 millimeters (negative 0.6 to 0.6 inch) at 0 to negative 20 and 20 millimeters (negative 0.8 to 0.8 inch) at 300 seconds.

Figure 80. Lower end Y-motion, time history of setup 3A at U equals 22 meters per second (72 feet per second). Graph.

The graph shows the lower end Y-motion displacement for setup 3A with the wind velocity at 22 meters per second (72 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is Y-displacement at the lower end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement forms an almost perfectly straight line at 0 along the time continuum from 0 to 300 seconds.

Figure 81. Top end Y-motion, time history of setup 3A at U equals 22 meters per second (72 feet per second). Graph.

The graph shows the top end Y-motion displacement for setup 3A with the wind velocity at 22 meters per second (72 feet per second). The horizontal axis is time in seconds ranging from 0 to 300. The vertical axis is Y-displacement at the top end in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches). The displacement forms an almost perfectly straight line centering at negative 5 millimeters (negative 0.2 inch) along the time continuum from 0 to 300 seconds.

The graph shows the trajectory of setup 2A with wind velocity at 18 meters per second (59 feet per second) in the first 5 minutes. The horizontal axis is the mean X-displacement in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches) while the vertical axis is the mean Y-displacement ranging from negative 50 to 50 millimeters (negative 2 to 2 inches). The plot shows a saucer-like ellipse centered on negative 10 millimeters (negative 0.4 inch) with the larger X-displacement around negative 50 to 45 millimeters (negative 2.0 to 1.8 inch) and the Y-displacement at negative 10 to negative 5 millimeters (negative 0.4 to negative 0.2 inch).

The graph shows the trajectory of setup 2A with wind velocity at 18 meters per second (59 feet per second) in the second five minutes. The horizontal axis is the mean X-displacement in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches) while the vertical axis is the mean Y-displacement ranging from negative 50 to 50 millimeters (negative 2 to 2 inches). The plot shows a saucer-like ellipse centered on negative 10 millimeters (negative 0.4 inch) with the larger X-displacement at negative 60 to 55 millimeters (negative 2.3 to 2.1 inch) and the Y-displacement at negative 10 to negative 5 millimeters (negative 0.4 to negative 0.2 inch).

The graph shows the trajectory of setup 2A with wind velocity at 19 meters per second (62 feet per second). The horizontal axis is the mean X-displacement in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches) while the vertical axis is the mean Y-displacement ranging from negative 50 to 50 millimeters (negative 2 to 2 inches). The plot shows a flat saucer-like ellipse centered on negative 10 millimeters (negative 0.4 inch) with the larger X-displacement at negative 70 to 60 millimeters (negative 2.7 to 2.3 inches) and the Y-displacement at negative 10 to negative 5 millimeters (negative 0.4 to negative 0.2 inch).

The graph shows the trajectory of setup 1B with wind velocity at 24 meters per second (79 feet per second). The horizontal axis is the mean X-displacement in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches) while the vertical axis is the mean Y-displacement ranging from negative 50 to 50 millimeters (negative 2 to 2 inches). The plot shows a flat saucer-like ellipse centered on 5 millimeters (0.2 inch) with the larger X-displacement at negative 35 to 30 millimeters (negative 1.4 to 1.2 inch) and the Y-displacement around 3 to 5 millimeters (0.1 to 0.2 inch).

The graph shows the trajectory of setup 1C with wind velocity at 36 meters per second (119 feet per second). The horizontal axis is the mean X-displacement in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches) while the vertical axis is the mean Y-displacement ranging from negative 50 to 50 millimeters (negative 2 to 2 inches). The plot shows an upright ellipse centered around 10 millimeters (0.4 inch) with the X-displacement 5 to 15 millimeters (0.2 to 0.6 inch) and the larger Y-displacement at negative 23 to 25 millimeters (negative 0.9 to 1.0 inch).

The graph shows the trajectory of setup 3A with wind velocity at 22 meters per second (72 feet per second). The horizontal axis is the mean X-displacement in millimeters ranging from negative 100 to 100 (negative 3.9 to 3.9 inches) while the horizontal axis is the mean Y-displacement ranging from negative 50 to 50 millimeters (negative 2 to 2 inches). The plot shows a flat saucer-like ellipse centered on negative 10 millimeters (negative 0.4 inch) with the larger X-displacement at negative 20 to 20 millimeters (negative 0.8 to 0.8 inch) and the Y-displacement negative 5 to 0 millimeters (negative 0.2 to 0 inch).

This graph compares the divergent response of the sway and vertical amplitude for setup 2A. The horizontal axis is the wind velocity (U) in meters per second ranging from 0 to 40 (0 to 131 feet per second). The vertical axis is amplitude (the minimum subtracted from the maximum and divided by 2 to derive an average) in millimeters. The vertical amplitude stays below 10 millimeters (0.4 inch) from 10 to 40 meters per second (33 to 131 feet per second). The sway peaks at near 70 millimeters (2.7 inches) at 19 wind velocity but falls quickly to below 10 millimeters (0.4 inch) except for two points at 15 millimeters (0.6 inch) near the 35 to 40 meters per second (115 to 131 feet per second) velocity.

This graph compares the divergent response of the sway and vertical amplitude for setup 1B. The horizontal axis is the wind velocity (U) in meters per second ranging from 0 to 40 (0 to 131 feet per second). The vertical axis is amplitude (the minimum subtracted from the maximum and divided by 2 to derive an average) in millimeters. The vertical amplitude stays below 5 millimeters (0.2 inch) from 7 to 37 meters per second (23 to 121 feet per second). The sway peaks at near 32 millimeters (1.2 inches) at 24 and 26 wind velocity but falls quickly to 5 millimeters (0.2 inch) at 28 meters per second (92 feet per second) velocity.

This graph compares the divergent response of the sway and vertical amplitude for setup 1C. The horizontal axis is the wind velocity (U) in meters per second ranging from 0 to 40 (0 to 131 feet per second). The vertical axis is amplitude (the minimum subtracted from the maximum and divided by 2 to derive an average) in millimeters. The vertical amplitude stays below 5 millimeters (0.2 inch) except for two peaks that rise to near 10 millimeters (0.4 inch) at 33 and 37 meters per second (108 and 121 feet per second). The sway peaks at near 25 millimeters (1.0 inch) at 37 meters per second (121 feet per second) wind velocity but falls back to 10 millimeters (0.4 inch) at near 40 meters per second (131 feet per second) velocity.

This graph compares the divergent response of the sway and vertical amplitude for setup 3A. The horizontal axis is the wind velocity (U) in meters per second ranging from 0 to 40 (0 to 131 feet per second). The vertical axis is amplitude (the minimum subtracted from the maximum and divided by 2 to derive an average) in millimeters. The vertical amplitude stays below 5 millimeters (0.2 inch) from 16 to 25 meters per second (52 to 82 feet per second). The sway peaks at near 21 millimeters (0.8 inch) at 22 meters per second (72 feet per second) wind velocity but falls quickly to 6 millimeters (0.2 inch) near 25 meters per second (82 feet per second) velocity.

This graph compares the divergent response of the sway and vertical amplitude for setup 3B. The horizontal axis is the wind velocity (U) in meters per second ranging from 0 to 45 (0 to 148 feet per second). The vertical axis is amplitude (the minimum subtracted from the maximum and divided by 2 to derive an average) in millimeters. Both the vertical and sway amplitude track below 5 millimeters (0.2 inch) from 8 to 34 meters per second (26 to 111.5 feet per second) velocity in a similar pattern.

The graph depicts a range of Reynolds numbers for smooth and sanded surface cables that predicts instability. V equals the velocity, lowercase D is the diameter of the cylinder, lowercase K is the equivalent roughness of the cable surface material, and the roughness ratio is defined by the equivalent cable surface roughness divided by the cylinder diameter. The horizontal axis is the Reynolds number ranging from 10 to the fourth power to 10 to the seventh power. The vertical axis is the drag coefficient, the constant C subscript lowercase D. The sanded cables show a lower range of 3 times 10 to the fourth power to 1 times 10 to the fifth power, while the smooth surface under uniform flow is 2 to 4 times 10 to the fifth power.

The graph shows plots of four different levels of damping. No additional damping was used for the low case, the intermediate used 16 elastic bands on each sway spring in the X-direction, the high damping used 28 elastic bands on each spring in both X and Y directions, and the very high damping level used the airpot dampers at both ends of the model. The horizontal axis is sway amplitude in millimeters ranging from 0 to 40 (0 to 1.6 inches). The vertical axis is damping in percent of critical ranging from 10 to the negative 2 to 10 to the 0 power. The low and intermediate cases achieve a sway amplitude of almost 40 millimeters (1.6 inches) at between 10 to the negative 2 and 10 to the negative 1 damping percent. The high damping with elastic bands shows the least sway amplitude at 17 at the 10 to the negative 1 damping percent, while the very high reads at 30 millimeters (1.2 inch) at between 10 to the negative 1 and 10 to the 0 power.

This graph shows the damping effect at 45 degrees inclination comparing mean wind speed and sway amplitude. The horizontal axis is mean wind speed in meters per second ranging from 0 to 40 meters per second (0 to 131 feet per second). The vertical axis is sway amplitude in millimeters ranging from 0 to 35 (0 to 1.4 inches). The high and very high amplitude show steady sway amplitude that stays below 7 millimeters (0.3 inch) no matter what the wind speed. The low damping shows a peak of 31 millimeters (1.2 inches) at 24 meters per second (79 feet per second) wind speed while the intermediate damping has peaks of 10 millimeters (0.4 inch) at 17 meters per second (56 feet per second), 16 millimeters (0.6 inch) at 25 meters per second (82 feet per second), and 17 millimeters (0.7 inch) at 40 meters per second (131 feet per second) mean wind speed. These results indicate that limited-amplitude unstable motion can be suppressed by increasing cable damping.

The graph plots both the sway and vertical amplitude for rough and smooth cables for this setup. The horizontal axis is wind velocity in meters per second. The vertical axis is amplitude (minimum subtracted from the maximum to derive the average) in millimeters. Both the vertical amplitudes track at below 5 millimeters (0.2 inch) from 7 to 39 meters per second (23 to 128 feet per second). Both the rough and smooth sway show more variation: the rough sway peaks at 19 millimeters (0.7 inch) at 18 meters per second (59 feet per second), while the smooth sway peaks at 20 millimeters (0.8 inch) at 22 meters per second (72 feet per second). The increase in cable surface roughness makes the unstable response range shift to a lower wind speed.

The graph plots both the sway and vertical amplitude for rough and smooth cables for this setup. The horizontal axis is wind velocity in meters per second. The vertical axis is amplitude (minimum subtracted from the maximum to derive the average) in millimeters. Both the vertical amplitudes track at below 5 millimeters (0.2 inch) from 7 to 39 meters per second (23 to 128 feet per second). Both the rough and smooth sway show more variation: the rough sway peaks at 43 millimeters (1.7 inches) at 30 meters per second (98 feet per second), while the smooth sway has 2 peaks of 31 millimeters (1.2 inches) at 24 and 26 meters per second (79 and 85 feet per second).

The graph plots both the sway and vertical amplitude for rough and smooth cables for this setup. The horizontal axis is wind velocity in meters per second. The vertical axis is amplitude (minimum subtracted from the maximum to derive the average) in millimeters. Both the vertical amplitudes track at below 5 millimeters (0.2 inch) from 10 to 39 meters per second (33 to 128 feet per second). Both the rough and smooth sway show more variation: the rough sway peaks at 50 millimeters (2 inches) at 32 meters per second (105 feet per second) while the smooth sway peaks at 68 millimeters (2.7 inches) at 19 meters per second (62 feet per second).

This graph compares the sway amplitudes of three different cases with varying frequencies used. The horizontal axis is sway amplitude in millimeters from 0 to 30 (0 to 1.2 inches). The vertical axis is damping ratio in percent from 0.00 to 0.50. Case III with the frequency ratio of 0.997 has a much lower damping ratio within the same amplitude ranges as the other cases and sustains this uniformity of less than 0.10 from 10 to 26 millimeters (0.4 to 1 inch) of sway amplitude.

This graph compares the sway amplitudes of three different cases with varying frequencies used. The horizontal axis is vertical amplitude in millimeters from 0 to 30 (0 to 1.2 inches). The vertical axis is damping ratio in percent from 0.00 to 0.50. The data show no organized pattern, but the full range of damping ratios are represented with most points falling below 10 millimeters (0.4 inch) of vertical amplitude.

The graph compares the wind velocity and amplitude data for three different frequency ratio cases. The horizontal axis is wind velocity from 0 to 45 in meters per second (0 to 148 feet per second). The vertical axis is amplitude in the X-direction in millimeters from 0.00 to 40.0 (0 to 1.6 inches). Cases I and II show small variances between 3 and 12 millimeters (0.1 and 0.5 inch) from 15 to 40 meters per second (49 to 131 feet per second) velocity. Case III shows a larger sway with a peak of 35 millimeters (1.4 inches) at 20 meters per second (66 feet per second) and another at 17 millimeters (0.7 inch) at 33 meters per second (108 feet per second).

The graph compares the wind velocity and amplitude data for three different frequency ratio cases. The horizontal axis is wind velocity from 0 to 45 in meters per second (0 to 148 feet per second). The vertical axis is amplitude in the Y-direction in millimeters from 0.00 to 40.0 (0 to 1.6 inches). All cases show minimal variance with the amplitude ranging from 2 to 5 millimeters (0.08 to 0.2 inch) from 17 to 39 meters per second (56 to 128 feet per second).

This graph plots six scenarios to compare wind velocity-damping relation of inclined dry cable. The horizontal axis is the Scruton Number (S subscript C equals M (mass of cable per unit length) times zeta (damping ratio) divided by rho (air density) times D (cable diameter) squared) ranging from 0 to 60. The vertical axis is reduced wind velocity (U lowercase R) ranging from 0 to 200. Saito’s rig test data defines the boundary for instability for rain/wind oscillations shown as a rising solid line from 45 to 80 velocity and a Scruton number of ranging from 3 to 50. Miyata’s plot point is 2 at 80 wind velocity. Setup 1B, 1C, 2C, 2A and 3A all cluster at Scruton numbers below 11 with the reduced wind velocity (defined by Equation 10) ranging from 80 to 160.

Figure 104. Chart. Taut cable with linear damper.

T represents the taut cable tension that bisects the diagram. M equals the mass per unit length of the cable, and C is the damping coefficient at the damper location. The lowercase L subscript 1 and lowercase L subscript 2 are the location of the damper. This diagram illustrates the variables used to solve the differential equation over the two cable segments on each side of the damper to find cable displacement.

In this graph, the horizontal axis is kappa, the nondimensional damping parameter or the range of normalized damping coefficients from 0 to 1 in tenth increments. The vertical axis is the damping ratio ranging from 0 to 0.6 for the first five modes of damper location 0.02. The curves all coincide with 0.50, the optimum damping ratio for a linear damper for one mode of vibration.

Figure 106. Chart. Cable with attached friction/viscous damper.

T represents the taut cable tension that bisects the diagram. M equals the mass per unit length of the cable, and C is the damping coefficient at the friction damper (F subscript O) location. The lowercase L subscript 1 and lowercase L subscript 2 are the location of the damper. This diagram illustrates the variables used to solve the force-velocity relationship for the damper with a friction threshold.

Figure 107. Chart. Force-velocity curve for friction/viscous damper.

The diagram is an illustration of the geometry involved in equation 45. The basis is an X/Y grid. The horizontal axis is V, the velocity of the cable at the damper attachment point, F subscript D (the vertical axis) is the force of the damper, and F subscript O is the friction force indicated in two locations in the curve, from the positive abscissa upward and from the negative abscissa downward. The lines coming from the vertical axis form a slope of C against the horizontal axis.

Figure 108. Graph. Normalized damping ratio versus clamping ratio.

This graph plots a half-circle curve. The horizontal axis is the clamping ratio (theta subscript CI) ranging from 0 to 1 in tenth increments. The vertical axis is the normalized damping ratio from 0 to 0.6. The peak of the circle plots at the balance point of 0.5 damping ratio and 0.5 clamping ratio.

The graph compares the clamping ratio and the normalized viscous damper coefficient. The horizontal axis is the clamping ratio (theta subscript CI) ranging from 0 to 1 in tenth increments. The vertical axis is kappa, the normalized viscous damper coefficient, ranging from 0 to 1 in tenth increments. The plot is a curve that slowly ascends to the 0.2 damper coefficient until the clamping ratio reaches 0.8, where the curve ascends rapidly to 1.

Figure 110. Graph. Relationship between nondimensional parameters mu and kappa with different values of the clamping ratio theta subscript CI for a friction/viscous damper.

The graph compares various clamping ratios (theta values) to the damper coefficient (kappa) on the horizontal axis ranging from 0 to 1 in tenth increments to the viscosity of air (mu) ranging from 0 to 3 on the vertical axis. A clamping ratio of one represents a clamped limit where the cable is completely restrained at the damper location and plots as a straight line at 2.5 viscosity of air. All values radiate out from 2.5 with the theta of 0.5 the most ideal, which has a damper coefficient of 0.1 and a 1.75 viscosity of air value.

In this graph, the horizontal axis is kappa, the nondimensional damping parameter or the range of normalized damping coefficients from 0 to 1 in tenth increments. The vertical axis is the damping ratio ranging from 0 to 0.6. The curves have various viscosity of air values, ranging from 0 to 2.4. Mu equals 0 is the asymptotic solution for the linear viscous damper and the curve duplicated in figures 8 and 105. Increasing the friction shifts the optimal portion of the curve left off the grid.

In this graph, the horizontal axis is kappa, the nondimensional damping parameter or the range of normalized damping coefficients from 0 to 1 in tenth increments. The vertical axis is the damping ratio ranging from 0 to 0.6 for the first five modes of damper location 0.02. The curves all coincide with 0.50 the optimum damping ratio for a linear damper for one mode of vibration when beta (the yaw angle between the wind direction and longitudinal bridge axis) is 0.5.

In this graph, the horizontal axis is mode ratio ranging from 1 to 10. The vertical axis is the damping ratio ranging from 0 to 0.6 when beta (the yaw angle between the wind direction and longitudinal bridge axis) is 1. The curve peaks at a damping ratio of 0.5 when the mode is 1 and decreases to a damping ratio of 0.10 when the mode is 10.

In this graph, the horizontal axis is amplitude ratio ranging from 1 to 10. The vertical axis is the damping ratio ranging from 0 to 0.6 when beta (the yaw angle between the wind direction and longitudinal bridge axis) is 0.5. The curve peaks at a damping ratio of 0.5 when the amplitude is 1 and decreases more gently than the mode ratio graph to a damping ratio of 0.25 when the mode is 10.

In this graph, the horizontal axis is mode-amplitude ratio ranging from 1 to 10. The vertical axis is the damping ratio ranging from 0 to 0.6 when beta (the yaw angle between the wind direction and longitudinal bridge axis) is 0. The curve rises steeply to a peak of 0.5 damping ratio when the mode-amplitude is 0.75 and decreases more rapidly than the mode ratio graph to a damping ratio of 0.075 when the mode is 10.

Figure 116. Chart. General problem formulation.

The diagram shows a simplified network with a set of two taut cables (T subscript 1 and T subscript 2 cable tension) that are L subscript 1 and L subscript 2 long, connected by a vertical rigid rod (P subscript 1 and P subscript 2). Cable tensions are divided into Elements 1,1; 1,2 and 2,1; 2,2. The location of the vertical rigid connection crosstie is represented by I subscript 1 and I subscript 2. The horizontal and vertical coordinates are represented by X and Y, respectively, with subscripts 11, 12, 21, and 22 representing the origin located at the left end of the top cable, the right end of the top cable, the left end of the bottom cable, and the right end of the bottom cable, respectively.

The free-vibration analysis method was used to study a cable network modeled after the Fred Hartman Bridge. The main south tower span unit shows a set of 12 stays, numbered from 13S to 24S with the latter as the reference element. The three transverse connectors (an eight-loop steel wire rope system) labeled Restraint 1 through 3 are spaced about one-fourth the distance along the reference cable 24S.

The eigenfunctions for the model show that, along a normalized abscissa on the first graph (mode 1), the whole set of cables is involved uniformly in the oscillation for an even pattern, while as the frequency increases each graph becomes progressively more out of alignment with an uneven amplitude pattern influenced by the location of the transverse connectors, governed by the distance between consecutive connectors. Mode 8 shows considerable distortion in both middle traverse sections.

This graph shows the natural frequencies plotted as a function of the mode number and compared to individual cable behavior. The horizontal axis is the mode number from 0 to 35. The vertical axis is frequency in hertz, ranging from 0.5 to 4.5. The lower limit of the plateau frequency interval is 1.9, while the upper limit is 2.7 hertz. The graph shows a high density pattern from 0 to 5 modes up to 4.5 hertz frequency for the localized modes. The three comparative cases for network vibration are the original configuration, rigid transverse links and modified non-rigid configuration with ground restrainers that essentially overlap in the graph. The consecutive step pattern changes at the lower and upper frequency interval plateaus (1.9 and 2.7 hertz).

Figure 120. Chart. Generalized cable network configuration.

The graphic shows a cable network configuration with a set of parallel cables interconnected with restrainers. The cable length is lowercase L subscript lowercase J with lowercase P the cable segments divided by the transverse crossties. The reference abscissa with X- and Y-axes is also illustrated. The constitutive law for these connectors is simulated by linear springs K or a viscous damper lowercase C.

Figure 121. Chart. Twin cable with variable position connector.

This graphic shows a symmetric twin cable arrangement. The cable length is L, with T the cable tension and mu the cable mass. The segments are divided into four elements defined by the transverse cable P subscript 1 and P subscript 2. The X- and Y-axes are also illustrated. Case A is the linear spring model K, while case B shows the viscous damper lowercase C. This illustration follows a typical hyperbolic law where the nodes P subscript 1 and P subscript 2 and elements 1 and 3 are at rest, while segments 2 and 4 are vibrating out of phase.

This 3-dimensional graph shows a frequency solution for a linear spring model with the modal frequency being a function of the normalized stiffness parameter and the normalized connector location. The two independent variables, represented by two mutually perpendicular horizontal axes, are the normalized stiffness parameter, denoted by lowercase D subscript K, and the normalized connector location, denoted by the Greek letter ?? The dependent variable, represented by the vertical axis, is the real reduced frequency ranging from 1 to 2 hertz. The segments wrap around the cubic graph and show a peak of 1.8 hertz reduced frequency.

Figure 123. Graph. Typical solution curves of the complex frequency for the dashpot.

The graph depicts the damping ratio of the first five PS modes with location C equal to 0.35. The horizontal axis is the normalized frequency. The vertical axis is damping ratio from 0 to 0.8. Modes 1 and 4 are slight curved lines to the right moving off the upper scale toward critical damping. Mode 3 is a dot on the horizontal axis, and modes 2 and 5 form half ellipse curves peaking at 0.3 and 0.1 percent damping ratio, respectively. The diamond symbols, representing the case of perfectly rigid restrainer, fall on the beginning of the mode 2 and mode 5 curves and on the end of the mode 3 curve.

Figure 124. Chart. Intermediate segments of specific cables only.

In this graphic that depicts a rigid restrainer, segment 1 is an unbroken line that forms a perfect square, and segment two is a wavy line bowed out at the top and bottom showing it vibrating out of phase with wavelength L to lowercase L. The associated formula is omega equals 1 divided by 1 minus XI times omega, where rho or air density is moving toward infinity.

Figure 125. Chart. Fred Hartman Bridge (A-line) 3D network.

This graphic shows a modification of the original system shown for the Fred Hartman Bridge. Stays AS1 through AS12 are shown with AS1 the reference. In addition to the three crosstie restraints, damper locations are at AS6 and AS10.

Figure 126. Chart. Equivalent model.

The equivalent model for the Fred Hartman Bridge shows the cable pattern in 2 dimensions. The bridge cables broken into segments by the restrainers are labeled Element 1, P. On the reference cable, restrainer 1 is located at 43.4 meters (142.3 feet), restrainer 2 at 44.6 meters (146.3 feet) and restrainer 3 at 41.1 meters (134.8 feet). The dampers directly connected to the bridge deck are D1 connected at restrainer 3 and D2 connected at restrainer 2.

The graph shows frequency solutions in the first mode for damped cable. The horizontal axis is normalized frequency from 1.00 to 2.00. The vertical axis is damping ratio from 0.00 to 0.25. With damper one only, the half circle ellipse starts at 1.45 and peaks at the optimized solution of 0.05 damping ratio and ends at 1.55 hertz frequency. Adding damper 2, represented by the thick line, the ratio goes from the first damper optimal peak to both dampers optimal at 0.23 damping ratio and then descends to 1.97 hertz frequency. The location of D2 at midspan is clearly more efficient.

This graph shows the complex eigenfunction with the first mode of the optimized system at 6.25 set against a horizontal axis of a normalized abscissa that ranges from 0 to 1. The plot shows a combination of in-phase and out-of-phase components when omega equals 0.97 hertz.

The graph shows the damping ratios for both stays for the first 10 vibration modes, computed from the eigenvalue equations along with corresponding curves predicted by the asymptotic expression for the universal curves for these values. The horizontal axis is the mode number, lowercase I, from 1 to 10 with the vertical axis the damping ratio, zeta, from 0.0 to 2.5 percent. AS23 forms a curve that descends from 1.9 at 0 to 0.4 percent damping ratio for mode 10. The AS16 stay forms a similar curve at higher values starting at 2.4 at 0 to 0.5 percent damping ratio at mode 10. The greatest discrepancy between the numeric and asymptotic values occurs at the optimal damper portion of the curve. The universal curves give good predictions at the higher modes because the damper ratio is not as optimal.

The six graphs show the global performance of stay AS16 with in-plane oscillations as a function of wind speed and direction for pre- and post-damper installation and damper force as a function of deck-level wind velocity and wind direction. The horizontal axis is either 1-minute mean deck-level wind direction in degrees or wind speed from 0 to 20 meters per second (0 to 66 feet per second). The vertical axis is either 1-minute RMS acceleration from 0 to 4 or RMS force from 0 to 6 in kilonewtons (0 to 1,350 poundforce). Before damper installation, a high density of points clusters near the abscissa from vortex-induced vibration and low-level buffeting response. Wind/rain oscillation causes multiple points over a wide wind speed with amplitudes as high as 2 and 2.5. The wind orientation cluster focuses in the 45- to 180-degree direction with highs in the 90- to 140-degree orientations. Post-damper the amplitudes tend to flatten out with more wind speeds present but lower RMS acceleration amplitudes that rarely exceed 0.5. The wind direction has also evened out to have more representation in the 225- to 315-degree range. The damper force also clusters in below 2 in the 0 to 10 meters per second (0 to 33 feet per second) wind range with some representation of forces of 4 and 5 in the 5 to 10 meters per second (16 to 33 feet per second) range. For the wind directions correlating with damper force, the directions tend to cluster at 45 to 135 degrees and 270 to 315 degrees below 2 kilonewtons (450 poundforce) with high forces up to between 4 and 5 kilonewtons (900 and 1,125 poundforce) at 90 to 135 and 225 to 270 degrees, which shows the dampers are functioning to provide dissipation of force to the stays.

The six graphs show the global performance of stay AS23 with in-plane oscillations as a function of wind speed and direction for pre- and post-damper installation and damper force as a function of deck-level wind velocity and wind direction. The horizontal axis is either 1-minute mean deck-level wind direction in degrees or wind speed from 0 to 20 meters per second (0 to 66 feet per second). The vertical axis is either 1-minute RMS acceleration from 0 to 4 or RMS force from 0 to 6 in kilonewtons (0 to 1,350 poundforce). Before damper installation, a high density of points clusters near the abscissa from vortex-induced vibration and low-level buffeting response. Wind/rain oscillation causes multiple points over a wide wind speed with amplitudes as high as 2.5 and 3. The wind orientation cluster focuses in the 25- to 100-degree direction with highs in the 90- to 135-degree orientations. Post-damper the amplitudes tend to flatten out with more wind speeds present but lower RMS acceleration amplitudes that rarely exceed 0.5. The wind direction has also evened out to have more representation in the 225- to 315-degree range. The damper force also clusters in below 2 in the 0 to 10 meters per second (0 to 33 feet per second) wind range with some representation of forces of 3 and 5 in the 3 to 10 meters per second (10 to 33 feet per second) range. For the wind directions correlating with damper force, the directions tend to cluster at 45 to 135 degrees and 270 to 360 degrees below 2 kilonewtons (450 poundforce) with high forces up to between 3 and 4 kilonewtons (675 and 900 poundforce) at 60 to 120 degrees, which shows the dampers are functioning to provide dissipation of force to the stays.

The two graphs show plots of 1-minute RMS displacement amplitude before dampers were installed for both the in-plane and lateral directions for two different stays. The horizontal axis is lateral displacement amplitude in centimeters on a scale of 0 to 10 (0 to 4 inches) for AS16 and 0 to 20 (0 to 8 inches) for AS23. The vertical axis is in-plane displacement amplitude in centimeters on a scale of 0 to 10 (0 to 4 inches). The left graph depicts AS16, which shows cluster A that corresponds to mode 1, cluster B that represents vibrations in the second mode, and cluster C that encompasses more than one mode. For stay AS23, all the points scatter from the 0,0 corner to 5 lateral displacement and near 20 in-plane displacement with no clear cluster pattern.

Figure 133. Chart. Sample Lissajous plots of displacement for two records from AS16.

These graphs are a breakdown of the two different records from stay AS16 to isolate the cluster patterns. The horizontal axis is lateral displacement in centimeters from negative 10 to 10 (negative 4 to 4 inches) for record 1 and negative 15 to 15 (negative 5.9 to 5.9 inches) for record 2. The vertical axis is in-plane displacement in centimeters from negative 10 to 10 (negative 4 to 4 inches). Record 1 from cluster A shows an almost straight line from 0 lateral placement running vertically from negative 10 to 10 centimeters (negative 4 to 4 inches) along the in-plane displacement scale. Record 2 from cluster C shows a more complex double ellipse crossing lines at 0 centimeter (0 inch) lateral displacement and 5 centimeters (2 inches) in-plane displacement. Although the in-plane displacement amplitudes are similar for both records, the lateral displacement amplitude for record 2 is significantly larger.

Figure 134. Chart. Power spectral density of displacement for two records from AS16.

These graphs show the power spectral density of records 1 and 2 for stay AS16. The horizontal axis is frequency in hertz from 0 to 20. For record 1, the vertical axis is PSD of in-plane displacement ranging from 0 to 150 while the lateral displacement ranges from 0 to 0.2. The PSD of in-plane displacement spikes to near 150 at 2 hertz. While the PSD of the lateral displacement also spikes at 2 hertz, that reading is less than 0.2. For record 2 the PSD in-plane displacement scale is from 0 to 100. There are three spikes at less than 5 hertz that read 100, 45, and 15. The PSD of the lateral displacement scale is 0 to 200, and the spikes at less than 5 hertz read at 175, 40, and 10. Vibrations in record 1 are predominantly mode 1, while record 2 has contributions from the first three modes.

Figure 135. Graph. Sample Lissajous plots of displacement for two records from AS23.

These graphs are a breakdown of the two different records from stay AS23. The horizontal axis is lateral displacement in centimeters from negative 20 to 20 (negative 8 to 8 inches). The vertical axis is in-plane displacement in centimeters from negative 20 to 20 (negative 8 to 8 inches). Record 1 shows an almost straight line from 0 lateral displacement running vertically from negative 20 to 20 centimeters (negative 8 to 8 inches) along the in-plane displacement scale. Record 2 shows an ellipse that has a lateral displacement of negative 10 centimeters (negative 4 inches) and an in-plane displacement of 20 centimeters (8 inches) at one end and 15 centimeters (5.9 inches) lateral displacement and negative 20 centimeters (8 inches) in-plane displacement at the other end. Record 2 is highly 2-dimensional, but still dominated by mode 3.

Figure 136. Graph. Power spectral density of displacement for two records from AS23.

These graphs show the power spectral density of records 1 and 2 for stay AS23. The horizontal axis is frequency in hertz from 0 to 20. For record 1, the vertical axis is PSD of in-plane displacement ranging from 0 to 1,000 while the lateral displacement ranges from 0 to 1.5. The PSD of in-plane displacement spikes to near 900 at 2 hertz. While the PSD of the lateral displacement also spikes at 2 hertz, that reading is 1.4 with two minor readings at near 0. For record 2, the PSD in-plane displacement scale is also 0 to 1,000, but the lateral displacement scale only ranges from 0 to 400. The spike at 2 hertz is about 375 in PSD lateral displacement. Most 2-dimensional vibrations for stay AS23 have multiple modes.

The two graphs show plots of 1-minute RMS displacement amplitude after dampers were installed for both the in-plane and lateral directions for two different stays. The horizontal axis is lateral displacement amplitude in centimeters on a scale of 0 to 10 (0 to 4 inches) for AS16 and 0 to 20 (0 to 8 inches) for AS23. The vertical axis is in-plane displacement amplitude in centimeters on a scale of 0 to 10 (0 to 4 inches) for AS 16 and 0 to 20 centimeters (0 to 8 inches) for AS23. The left graph shows AS16 has in-plane and lateral displacement amplitudes of less than 3, while AS23 amplitudes are clustered in the 0,0 corner. Few records exceed 0.5 centimeters (0.2 inch) which confirms that dampers are effective in suppressing large-amplitude vibrations.

Figure 138. Graph. Lissajous and power spectral density plots of displacement for record A.

These graphs show the Lissajous and power spectral density plots for record A of stay AS16. For the Lissajous plot, the horizontal axis is lateral displacement in centimeters ranging between negative 6 and 6 (negative 2.3 and 2.3 inches). The vertical axis is in-plane displacement in centimeters from negative 5 to 5 (negative 2 to 2 inches). The plot is a saucer-shape almost on a straight horizontal line with 0. In the second graph, the horizontal axis of the PSD in-plane displacement ranges from 0 to 20. The vertical axis of the PSD in-plane displacement ranges from 0 to 0.2, with two smaller spikes at 0.03 at less than 5 hertz frequency. The largest spike at 3 hertz gives a reading of 0.17. In the third graph, the PSD of the lateral displacement has a vertical axis ranging from 0 to 25. All spikes are below 5 hertz, with the largest reading at 21 and smaller spikes at 5 and 9 PSD of lateral displacement. Although the vibration is basically 1-dimensional, the lateral displacement has contribution from all of the first three modes.

Figure 139. Graph. Modal frequencies of stays (A) AS16 and (B) AS23.

The graphs compare the modal frequencies of the two stays with and without the dampers. The horizontal axis is the mode number from 0 to 10. The vertical axis is the frequency in hertz from 0 to 10. The lower modes have lower frequencies. AS16 starts at 1.5 hertz at 1 and climbs to 9 hertz at mode 7. For AS23 the frequency starts at 0.25 hertz at mode 1 and climbs less steeply to 5.5 hertz at mode 8. For both stays, the with damper and without damper lines nearly coincide.

The estimated second-mode frequency of stay ASA16 without a damper is plotted against the RMS displacement at the accelerometer location. The horizontal axis is RMS displacement in centimeters ranging from 0 to 8 (0 to 3.1 inches). The vertical axis is second mode frequency in hertz from 0 to 2.6. For the plot without damper, the points scatter from 0 to 6 centimeters (0 to 2.3 inches) at below 2.5 hertz. The plot with a damper shows a greater concentration of points at a higher frequency between 2.55 and 2.6 hertz, but the displacement does not exceed 2 centimeters (0.8 inch).

The two graphs show the effect of the attached damper on stay AS16. The horizontal axis is the mode number from 0 to 8, and the vertical axis is damping ratio in percent from 0 to 1 in tenth increments. Without rain, the curve starts at 0.8 at one and decreases to 0.1 percent at mode 7. Mode 1 is the only value where with and without damper differ, because the damping ratio is 0.65 percent before and 0.8 with the damper. The largest increase is 0.15 percent in the first vibration mode without rainfall. In the second graph, the damping ratios start lower at near 0.7 percent and fall to 0.15 at mode 7. The largest increase of 0.06 percent comes in the second mode, the most dominant mode for wind-rain induced vibration.

The bar graphs show frequency of damping ratios from records without rainfall before the dampers were installed. For AS16 in mode 2, the horizontal axis is damping ratio in percent from 0.2 to 0.6, while the range for AS23 in mode 3 is 0.1 to 0.8 percent. The vertical axis is occurrence frequency from 0 to 0.5. AS16 shows a more even distribution, with 0.3 occurrence frequency at 0.35 and 0.4 damping ratio at 0.3. For AS23, all values have an occurrence frequency of 0.2 or less, except for the 0.5 damping ratio that shows a 0.475 occurrence frequency.

Figure 143. Graph. Dependence of modal damping on damper force.

This graph shows the relation between estimated damping in the second mode of stay AS16 with corresponding damper force. The horizontal axis is damper force in kilonewtons ranging from 0 to 5 (0 to 1,125 poundforce). The vertical axis is damping ratio in the second mode ranging from 0 to 0.9 in tenth increments. The plot includes with and without rain and the points are scattered from 0.3 to 0.55 for the range of damper force from 0 to 4.5 kilonewtons (0 to 1,012.5 poundforce). The trend is that the modal damping ratio decreases when the damper is more engaged with rain.

The graphs measure the damper force for cables AS16 and AS23. The horizontal axis is RMS displacement at damper-cable connection ranging from 0 to 2 centimeters (0 to 0.8 inch),while the vertical axis is RMS displacement at the accelerometer location ranging from 0 to 2 centimeters (0 to 0.8 inch). For AS16, the majority of the points cluster along the vertical axis with no point more than 0.25 centimeter (0.1 inch) displacement at the damper cable. All records had one-minute RMS damper force greater than 0.5 kilonewtons (112.5 poundforce). The four clusters represent in-plane vibrations for the second, fourth, fifth, and seventh modes. Cable AS23 shows a more prominent clustering at below 0.5 accelerometer location, but no predominant modes. The graphs suggest that the damper may have been frequently locked up and not producing energy dissipation.

Figure 145. Graph. Damper force versus displacement and velocity for a segment of a sample record.

These graphs plot the measured damper force against the displacement and velocity. The vertical axis is damper force from negative 5 to 5 kilonewtons (negative 1,125 to 1,125 poundforce). In the first graph, where the horizontal axis is integrated displacement in centimeters from negative 0.1 to 0.1 (negative 0.04 to 0.04 inch), the reading is a square pattern in the middle of the graph between negative 0.05 and 0.05 centimeters (negative 0.02 and 0.02 inch) displacement and negative 5 to 5 kilonewtons (negative 1,125 to 1,125 poundforce) of damper force. In the second graph, the vertical axis is integrated velocity in centimeters per second ranging from negative 1.5 to 1.5 (negative 0.6 to 0.6 inch). The readings form an S-shape centered in the middle with ends at negative 5 kilonewtons (negative 1,125 poundforce), negative 1 centimeters per second (negative 0.4 inches per second), and 5 kilonewtons force (1,125 poundforce) and 0.75 centimeters per second (0.3 inches per second) velocity. The measured displacement shows a similar square plot to the integrated displacement and the differentiated velocity shows a more disparate S-shape that the integrated velocity. These abrupt changes in the curvatures of the force-displacement plots and the slopes of the lines in the force-velocity plots suggest the damper did not behave viscously. Instead, it locked up until the damper force passed a certain level which turns out to be the residual static friction of the damper.

Figure 146. Chart. Displacement and damper force time histories of a sample record.

The two graphs show time histories of displacement and damper force for AS16. The horizontal axis is damper time in seconds from 0 to 300. The first plot is integrated displacement in centimeters ranging from negative 2 to 2 (negative 0.8 to 0.8 inch). The plot shows peaks of near negative 2 and 2 centimeters (negative 0.8 to 0.8 inch) at 50 seconds that level off to about negative 1.5 and 1.5 (negative 0.6 and 0.6 inch) for the rest of the time period. The second graph shows the vertical axis is damper force from negative 5 to 5 kilonewtons (negative 1,125 to 1,125 poundforce). Again, the plot shows a peak at about 50 seconds of negative 6 to 6 kilonewtons (negative 1,350 to 1,350 poundforce) and then levels off to less than negative 5 to 5 kilonewtons (negative 1,125 to 1,125 poundforce). The periodic increasing and decreasing of amplitudes at the same level suggests the damper keeps the vibration below a certain level.

Figure 147. Drawing. Incline stay cable properties.

The diagram illustrates a typical stay cable and the variables that represent the dimensions and angles to solve cable mechanical problems, showing a triangle with the cable sway between the low and high points and extended vertical and horizontal lines for additional angles. T subscript 0 is the lower cable tension, and T subscript 1 is the upper cable tension in kilonewtons or kips. V and H are the vertical and horizontal components of the cable tension in kilonewtons or kips with corresponding Y and X dimensions. L equals the cable run in meters or feet, and H equals the cable rise in meters or feet. Theta is the cable angle.

Figure 148. Drawing. Definition diagram for a horizontal cable (taut string) compared to the definition diagram for an inclined cable.

This figure shows two drawings to describe the dynamic cable properties. The horizontal cable (taut string) shows T, a curved line of cable tension with H, the horizontal plane; A and B endpoints with the X dimension that equals L; and Z, the distance from the horizontal line X and the midpoint of the curved T line. The inclined cable shows the L dimension on an angle, theta. The H line forms a right triangle with the X dimension and the X tangent of theta and the Z dimension from the H asterisk to the AB curved line.

The figure plots the variable, defined by the square root of T divided by M, versus unstressed cable length for three bridges. The horizontal axis is unstressed cable length in meters ranging from 0 to 250 (0 to 820 feet) and the vertical axis is the square root of T divided by lowercase M, ranging from 0 to 300. All cable lengths start at 50 meters (164 feet) and extend to 250 meters (820 feet), with the square root of T divided by lowercase M between 150 and 250. The Alex Fraser, a helical twist parallel wire cable bridge, plots the highest, with an average of 231 for the square root of T divided by lowercase M, while Maysville plots the lowest with an average of 188.

The graph shows the first two modes of the unstressed cable length and the cable frequency for the three bridges. The horizontal axis is unstressed cable length in meters ranging from 49 to 229 (161 to 751 feet). The vertical axis is frequency in hertz from 0.0 to 6.0. For all three bridges, the mode 1 values are less than the corresponding mode 2 values. The lowest frequencies are 2 hertz for the 124-meter (407-foot) mode 1 cable on the Marysville Bridge and the 142-meter (466-foot) cable on the Owensboro Bridge. The highest value is 5 hertz for the 229 plus cable in mode 2 for the Alex Fraser Bridge.

Figure 151. Photo. RAMA 8 bridge (artistic rendering).

The RAMA Bridge in Bangkok, Thailand, is a cable-stayed bridge with four lanes of traffic and two pedestrian sidewalks on the main span. The sketch shows the 160-meter (525-foot) tower, an inverted concrete Y. There are two inclined cable-stay planes with 28 cables each for the main span and 1 cable-stay plane with 28 cables on the anchor span.

The computer model was created to study the global behavior of the bridge under static and dynamic loads. The model has 1,014 beam elements and 84 cable elements. The XZ view shows the bridge in profile with piers 44 through 40 to the bridge tower on the left of the drawing and piers 38 and 39 on the right of the drawing at deck level. The XY view is a top view that shows the length of the bridge with piers and the cable arrays at the tower on the right side. The YZ view shows the bridge in side profile with the cable arrays on both sides of the tower.

The diagram shows cable M26 with the nodes and cable elements divided into 10 segments. The cable diagonally bisects the drawing from the upper left corner to the lower right with equal distance between the cable segments.

Figure 154. Chart. Cable catenary.

In this drawing, X is the horizontal axis and Y is the vertical axis. T subscript H is the horizontal cable force and theta is the angle between the cable and the horizontal deck. This drawing is the basis to use the perfect curve model equations to define cable nodes.

The figure shows three cable modes with their in-plane and out-of plane (perpendicular to its plane) vibrations. As the modes increase, the vibration pattern also becomes more complex with 2 and 3 loops. The out-of-plane vibrations occur in the horizontal XY and the vertical ZY views. The in-plane vibrations occur along the cable length in the XZ view.

The inextensible cable increases the cable area to 100,000 millimeters squared (3,900 inches squared) that causes the cable to be very stiff axially and gives results closer to the theoretical model. Modes shapes are similar to the extensible cable model except for the in-plane mode 1, which shows vibration without a sine shape in the XZ view. The period for this mode is only 27 percent because of the large increase in induced cable tension. Because the cable is so stiff, small differential movement creates large tensile forces in the cable which increases the frequency. This effect occurs only in symmetric modes and decreases as the mode number increases.

This side view of the bridge with cable M26 shows the 10 segments clearly.

Figure 159. Chart. Fundamental bridge modes.

For the first vertical mode where T equals 3.397 seconds and the frequency is 0.294 hertz, the vibration effect undulates near the bridge deck and shows a small vertical offset by the tower. For the first longitudinal mode where T equals 2.774 seconds and frequency is 0.360 hertz, the cables vibrate throughout with more displacement of the tower. In the first transverse mode where T equals 2.562 and the frequency is 0.390 hertz, the cables and the tower show bowing out movement in the XY and YZ views.

Figure 160. Chart. Additional bridge modes.

For the second vertical mode where T equals 2.205 seconds and the frequency is 0.454 hertz, the cable vibration is an S shape near the bridge deck. For the second transverse mode where T equals 2.062 seconds and the frequency is 0.485 hertz, the tower vibrates minimally and the cables skew out to the right in the XZ view and upward in the XY view with some skew in the deck. For the first torsional mode where T equals 1.863 seconds and the frequency is 0.537 hertz, the bottom section of the cables vibrate and the bridge deck torques in the XZ view.

Figure 161. Chart. Four first modes of the cables: XY, YZ, and XZ views.

Higher modes tend to be spatial and move in more complicated 3D motions because of interaction with the deck and the tower. The first model shows the cable bowing out in 2 directions in views XY and YZ. The second model shows the cable bowing out to the right in views XY and YZ. The third model shows the cable bowing up in view XZ and the fourth model shows the cable bowing up and down in view XZ and to the left in view YZ.

Figure 162. Chart. Four second modes of the cables: XY, YZ, and XZ views.

Higher modes tend to be spatial and move in more complicated 3D motions because of interaction with the deck and the tower. These models make either S-curves (models 2 and 4) or double ellipses (models 1 and 3) in all views.

Figure 163. Chart. Four third modes of the cables: XY, YZ, and XZ views.

Higher modes tend to be spatial and move in more complicated 3D motions because of interaction with the deck and the tower. These models make either triple S-curves (models 1 and 4) or triple ellipses (models 2 and 3) in all views.

This figure is the model used for static live load analysis. Results were computed and tabulated for nodal deformations at nodes 178, 283, 338, 427, and 463; girder member effects at 1,071, 1,075, 1,141, 1,145, 1,211, and 1,215; and cable numbers M7, M14, M21, and M26.

Figure 165. Graph. RAMA 8 Bridge model damping versus frequency.

The graph compares the frequency and damping percent in 5 modes. The horizontal axis is frequency in hertz ranging from 0.00 to 1.20. The damping percent range is 0.0 to 2.5. The alpha and beta damping coefficients were calculated using the frequencies of the 1st and 15th bridge natural modes. The damping coefficients were set at 0.75 percent of critical. All the curves (vertical, longitudinal, and transverse bridge modes and the first and second cable modes) nearly coincide (the damping percent starts at 2.5 at 0.10 hertz and falls to 0.75 at 0.30 to 1.20 frequency), showing that the frequency range of interest gives an acceptable damping distribution.

The graphs are time histories for the static and dynamic loads when the train crosses the bridge. The horizontal scale is time in seconds ranging from 0 to 40. In the first graph, the vertical axis is the displacement ranging from negative 350 to 50. The largest dip measures negative 350 at 12 seconds. The train has left the bridge by 20 seconds and both the static and dynamic loads show displacement hovering around 0. The second graph shows the vertical axis as the velocity in millimeters per second ranging from negative 150 to 200 (negative 5.9 to 7.8 inches per second). Again, the largest change is between 7 and 17 seconds where the velocity dips to negative 110 millimeters per second (negative 4.3 inches per second) and then rises to nearly 150 millimeters per second (5.9 inches per second). In the last graph, the vertical axis is acceleration in millimeters per second squared. The most dramatic changes occur again between 7 and 17 seconds when the acceleration ranges from negative 100 to 200 millimeters per second squared (negative 3.9 and 7.8 inches per second squared) and back down to negative 200 (negative 7.8 inches per second squared). The dynamic displacement was only slightly larger than the static displacement.

The graph shows the time history for the static and dynamic bending moment. The horizontal axis is time in seconds from 0 to 40. The vertical axis is bending moment from negative 3,000 to 6,000 kilonewton-meters (negative 2214 to 4428 kilopoundforce-feet). The two values coincide for the most part and show most of the readings near 0, except for the train passing that crates readings of negative 1,500 kilonewton-meters (negative 1107 kilopoundforce-feet) at 7 seconds, a peak of near 5,000 kilonewton-meters (3690 kilopoundforce-feet) at 12 seconds, and another dip at 15 seconds to negative 1,000 kilonewton-meters (negative 738 kilopoundforce-feet). This figure shows little amplification from dynamic effects.

In this graph, the horizontal axis is time from 0 to 40 seconds and the vertical axis is cable tension in kilonewtons from negative 50 to 300 (negative 11,250 to 67,500 poundforce). The total cable force time diagram shows that the static and dynamic time histories have the same shape, but dynamic effects cause slightly larger maxima (225 kilonewtons (50,625 poundforce)) and minima (negative 25 kilonewtons (negative 5,625 poundforce)) at between 7 and 17 seconds.

The horizontal axis is time in seconds from 0 to 40 and the vertical axis is delta T1 in kilonewtons (cable tension). The readings start at 0 and peak at 12 kilonewtons (2,700 poundforce) at 12 seconds and then drop to negative 7 kilonewtons (negative 1,575 poundforce) at 17 seconds, but then vary between negative 5 to 5 kilonewtons (negative 1,125 to 1,125 poundforce) for the remaining time.

The horizontal axis is frequency in hertz from 0 to 2. The vertical axis is spectra density from 0 to 16. The two dominant peaks of about 14 in the cable tension spectrum occur at 0.3 hertz and 0.44 hertz. The first peak corresponds to the first vertical deck mode, and the second is a mixture of the second deck mode and the first cable modes (in-plane and out-of-plane).

The three-part graphs show displacement in three directions for cable M26 at the quarter, half and three-quarter span, the tower end and the deck end. The horizontal axis is time in seconds from 0 to 40. In the longitudinal graph, which is the horizontal direction along the bridge axis, the displacement (delta X) in millimeters peaks at 12 seconds at 70 millimeters (2.7 inches) for the quarter- and half-span cable node. For the lateral or the Y transverse displacement, the graph shows more continuous displacement with lower peaks. The average displacement is between negative 5 and 5 millimeters (negative 0.2 and 0.2 inches), with a negative 15-millimeter (negative 0.6-inch) reading at 12 seconds. The last graph is the vertical displacement or delta Z. In this graph, the deck shows the most displacement at negative 275 millimeters (10.7 inches) at 12 seconds.

Figure 172. Chart. Transformation from global coordinates along the cable.

The graphic shows the global X (horizontal) and Y (transverse) coordinates transformed into a dotted line drawing of local cable coordinates U (along the cable), V (normal to the cable), and W (at right angles to both U and V). The graphic also shows angles alpha and beta and projection lines eta and xi.

Figure 173. Chart. Local coordinate displacements of nodes of cable M26 (millimeters). Displacements are shown for three nodes of the cable: At ¼ span (closer to the tower), ½ span, and ¾ span (closer to the deck; train speed equals 80 kilometers per hour (50 miles per hour)).

The six graphs are circle plots with rings starting at 0 and increasing in 30-milimeter increments to 150. The three-quarter span mode nearest to the deck shows the most displacement at 100 and 115 degrees. The out-of-plane deformations show that the top of the cable moves up as the sag is reduced, while the bottom of the cable moves down with the deck. The center of the cable shows little movement since the deck deformation and sag reduction cancel each other.

The graphs form a chart of spectra movements for three locations (one-quarter span, one-half span and three-quarters span) along M26 and three directions: U (along the cable), V (perpendicular to the cable plane), and W (perpendicular to the cable in cable plane). The horizontal axes are frequency in hertz and the vertical axes are spectral density. The U displacements along the cable are predominantly from the first vertical mode of the deck. The V displacements out of the cable plane are predominantly from the first transverse mode of the structure. The W displacement perpendicular to the cable in the cable plane shows a peak of 10 to 12 at 0.43 hertz at all three locations.

This time history graph of cable M26 shows deck and cable end rotations as a result of changes in sag from increased or decreased load and the cable rotation from deck deflections. The horizontal axis is time from 0 to 40 seconds. The vertical axis is rotation in degrees from negative 0.5 to 0.6. At 12 seconds, the minimum deck rotation is negative 0.3, and the maximum cable end rotation is 0.15 degrees. The heavy solid line, which is the individual rotation components and the total relative rotation, has a reading of 0.5 degrees.

This time history graph of cable M21 shows deck and cable end rotations as a result of changes in sag from increased or decreased load and the cable rotation from deck deflections. The horizontal axis is time from 0 to 40 seconds. The vertical axis is rotation in degrees from negative 0.2 to 0.3. The heavy solid line, which is the individual rotation components and the total relative rotation, has a peak reading of 0.25 and a minimum reading of negative 0.2 degrees. The cable end rotation peaks at 0.2 degrees at 12 seconds. The minimum deck rotation is negative 0.15 at 10 seconds, and its maximum is 0.25 at 15 seconds. This disparity in results shows that the maximum relative rotation cannot be reliably approximated by adding the maxima of the individual components.

Figure 177. Graph. Effect of mode (constant amplitude and velocity).

The graph shows a pedestrian and driver line with mode shape on the horizontal axis ranging from 0 to 4 and user comfort ranging from 5 to 1. From modes 1 to 3, both driver and pedestrian comfort was between 4 and 3. This graphs shows that mode shape does not affect the either driver or pedestrian comfort.

Figure 178. Graph. Effect of velocity (constant amplitude).

In this graph, the horizontal axis is frequency ranging from 0 to 2.5. The vertical axis is user comfort, which ranges from 5 to 1. Frequency ranges from 0.5 to 2 for the pedestrian and driver lines, and user comfort increases from 2.5 to 4.5. Although an amplitude that was too high was chosen as the constant, users had strong reservations when frequencies were higher than 1 hertz.

Figure 179. Graph. Effect of amplitude (constant velocity.)

The graph shows the horizontal axis as amplitude from 0 to 2.5. The vertical axis is user comfort ranging from 5 to 1. User comfort is around 2 when the amplitude is 0.5 and dips to 3 or 3.5 when amplitude is 1. Although the graph shows that users were more comfortable with oscillations of 2 diameters than 1, this was dismissed as an anomaly caused by the number and order of videos that had been shown up to this point in the survey.

Equations

Equation 82

The equation reads S is a matrix. The first row consists of elements of the sine of open parentheses alpha times pi times xi subscript 1 close parentheses, minus the sine of open bracket alpha times pi open parentheses 1 minus xi subscript 1 close parentheses close bracket, zero, and zero. The second row consists of elements of zero, zero, the sine of open parentheses alpha times pi times the increment of xi close parentheses, and minus the sine of open bracket alpha times lowercase F times pi times open parentheses L subscript 2 over L subscript 1 all minus the increment of xi close parentheses, close bracket. The third row consists of the sine of open parentheses alpha times pi times xi subscript 1 close parentheses, zero, minus the sine of open parentheses alpha times lowercase F times pi times the increment of xi close parentheses, and zero. The fourth row consists of the cosine of open parentheses alpha times pi times xi subscript 1 close parentheses, the cosine of open bracket alpha times pi times open parentheses 1 minus xi subscript 1 close parentheses, close bracket, lowercase F over lowercase H times the cosine of open parentheses alpha lowercase F times pi times the increment of xi close parentheses, and lowercase F over lowercase H times the cosine of open bracket alpha times lowercase F times pi times open parentheses L subscript 2 over L subscript 1 all minus the increment of xi close parentheses, close bracket.

Equation 83

The equation reads lowercase P of alpha is equal to the sine of open parentheses alpha times pi close parentheses times open parentheses the cosine of open bracket alpha times lowercase F times pi open parentheses 2 times the increment of xi minus L subscript 2 over L subscript 1 close parentheses close bracket minus the cosine of open parentheses alpha times lowercase F times pi times L subscript 2 over L subscript 1 close parentheses, close parentheses plus lowercase H over lowercase F times the sine of open parentheses alpha times lowercase F times pi times L subscript 2 over L subscript 1 close parentheses times open parentheses cosine open parentheses 2 times alpha times pi times xi subscript 1 minus alpha times pi close parentheses minus the cosine of open parentheses alpha times pi close parentheses, close parentheses.

Equation 128

The equation reads lowercase F of open parentheses omega star close parentheses is equal to open bracket, open parentheses omega star close parentheses, open parentheses L star close parentheses all over the sum of the square root of H star divided by lowercase M divided by 2 close bracket minus 4 over open parentheses lambda star close parentheses squared times open bracket, open parentheses omega star close parentheses, open parentheses L star close parentheses all over the sum of the square root of H star divided by lowercase M divided by 2 close bracket superscript 3 minus the tangent of open bracket, open parentheses omega star close parentheses, open parentheses L star close parentheses all over the sum of the square root of H star divided by lowercase M divided by 2 close bracket.